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          <h1 class="post-title" itemprop="name headline">2019-1-23 尔</h1>
        

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        <h2 id="problem"><a href="#problem" class="headerlink" title="problem"></a>problem</h2><p>给一个长度为n的序列。</p>
<p>支持两个操作</p>
<p>1.给区间[L，R]第i项加上斐波那契数列的第i项。</p>
<p>2.查询区间[L，R]的异或和。</p>
<p>$n,m&lt;3e5$</p>
<a id="more"></a>
<h2 id="solution"><a href="#solution" class="headerlink" title="solution"></a>solution</h2><p> 这一题其实要考虑Fibonacci数列的两个性质:</p>
<p> (i) $\sum_{i=1}^{n} fib(i)=fib(n+2)−1$</p>
<p>(ii)令$S<em>i=S</em>{i−1}+S_{i−2}$, </p>
<p>其中$S_1=a,S_2=b$</p>
<p> 那么$S_i=bFib(i−1)+aFib(i−2)$</p>
<p>还有一个推导式$\sum_{i=1}^{n} S(i)=S(n+2)−S_2$</p>
<p> 那么就很好做了， 在线段树中， 我们要只要记数列的前两项就可以方便的对数列进行求和。</p>
<p>然后用一个标记永久化的线段树做一下就可以了。</p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> fst first</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> snd second</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> squ(x) ((LL)(x) * (x))</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> debug(...) fprintf(stderr, __VA_ARGS__)</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">typedef</span> <span class="keyword">long</span> <span class="keyword">long</span> LL;</span><br><span class="line"><span class="keyword">typedef</span> pair&lt;<span class="keyword">int</span>, <span class="keyword">int</span>&gt; pii;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">read</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	<span class="keyword">int</span> sum = <span class="number">0</span>, fg = <span class="number">1</span>; <span class="keyword">char</span> c = getchar();</span><br><span class="line">	<span class="keyword">for</span> (; !<span class="built_in">isdigit</span>(c); c = getchar()) <span class="keyword">if</span> (c == <span class="string">'-'</span>) fg = <span class="number">-1</span>;</span><br><span class="line">	<span class="keyword">for</span> (; <span class="built_in">isdigit</span>(c); c = getchar()) sum = (sum &lt;&lt; <span class="number">3</span>) + (sum &lt;&lt; <span class="number">1</span>) + (c ^ <span class="number">0x30</span>);</span><br><span class="line">	<span class="keyword">return</span> fg * sum;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> Maxn = <span class="number">3e5</span> + <span class="number">10</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> mod = <span class="number">1e9</span> + <span class="number">7</span>;</span><br><span class="line"><span class="keyword">bool</span> ok;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">add</span><span class="params">(<span class="keyword">int</span> A,<span class="keyword">int</span> B)</span></span>&#123; </span><br><span class="line">	<span class="keyword">if</span>(A + B &gt;= mod) <span class="keyword">return</span> A + B - mod;</span><br><span class="line">	<span class="keyword">if</span>(A + B &lt; <span class="number">0</span>) <span class="keyword">return</span> A + B + mod;</span><br><span class="line">	<span class="keyword">return</span> A + B;</span><br><span class="line">&#125;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">STR</span>&#123;</span></span><br><span class="line">	<span class="keyword">int</span> s1, s2, len, sum;</span><br><span class="line">	STR <span class="keyword">operator</span> + (<span class="keyword">const</span> STR &amp;t) <span class="keyword">const</span>&#123;</span><br><span class="line">		<span class="keyword">if</span>(len == <span class="number">1</span>) <span class="keyword">return</span> (STR)&#123;add(s1 , t.s1), s2, len, sum&#125;;</span><br><span class="line">		<span class="keyword">return</span> (STR)&#123;add(s1 , t.s1), add(s2 , t.s2), len, sum&#125;;</span><br><span class="line">	&#125;</span><br><span class="line">&#125;tre[Maxn &lt;&lt; <span class="number">2</span>];</span><br><span class="line"><span class="keyword">int</span> fib[Maxn &lt;&lt; <span class="number">2</span>], sum[Maxn], ls[Maxn &lt;&lt; <span class="number">2</span>], rs[Maxn &lt;&lt; <span class="number">2</span>], cnt;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">Sum</span><span class="params">(<span class="keyword">int</span> l ,<span class="keyword">int</span> r)</span> </span>&#123; <span class="keyword">return</span> add(sum[r], mod - sum[l - <span class="number">1</span>]);&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">Fib</span><span class="params">(<span class="keyword">int</span> l,<span class="keyword">int</span> r)</span></span>&#123;</span><br><span class="line">	<span class="keyword">return</span> add(fib[r + <span class="number">2</span>], mod - fib[l + <span class="number">1</span>]);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">init</span><span class="params">(<span class="keyword">int</span> n)</span></span>&#123;</span><br><span class="line">	fib[<span class="number">1</span>] = <span class="number">1</span>;fib[<span class="number">2</span>] = <span class="number">1</span>;</span><br><span class="line">	<span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">3</span>; i &lt;= (n &lt;&lt; <span class="number">1</span>); ++i) fib[i] = add(fib[i - <span class="number">1</span>], fib[i - <span class="number">2</span>]);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">calc</span><span class="params">(STR &amp;t,<span class="keyword">int</span> L, <span class="keyword">int</span> R)</span></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(t.s1 == <span class="number">0</span> &amp;&amp; t.s2 == <span class="number">0</span>) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">	<span class="keyword">if</span>(L &gt; R) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">	<span class="keyword">if</span>(L == R) <span class="keyword">return</span> t.s1;</span><br><span class="line">	<span class="keyword">if</span>(L + <span class="number">1</span> == R) <span class="keyword">return</span> add(t.s1, t.s2);</span><br><span class="line">	<span class="keyword">return</span> add(add(<span class="number">1l</span>l * t.s2 * fib[R - L + <span class="number">2</span>] % mod ,mod - t.s2), <span class="number">1l</span>l * t.s1 * fib[R - L + <span class="number">1</span>] % mod);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">Sib</span><span class="params">(STR &amp;t,<span class="keyword">int</span> L, <span class="keyword">int</span> R, <span class="keyword">int</span> B)</span></span>&#123;</span><br><span class="line">	<span class="keyword">int</span> S_R = calc(t, B, R);</span><br><span class="line">	<span class="keyword">int</span> S_L = calc(t, B, L - <span class="number">1</span>);</span><br><span class="line">	<span class="keyword">return</span> S_R - S_L;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">modify</span><span class="params">(<span class="keyword">int</span> &amp;rt, <span class="keyword">int</span> l, <span class="keyword">int</span> r, <span class="keyword">int</span> L, <span class="keyword">int</span> R, <span class="keyword">int</span> B)</span></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(!rt) rt = ++cnt;</span><br><span class="line">	<span class="keyword">if</span>(l == L &amp;&amp; r == R) &#123;</span><br><span class="line">		tre[rt].len = r - l + <span class="number">1</span>;</span><br><span class="line">		tre[rt] = tre[rt] + (STR)&#123;fib[B],fib[B + <span class="number">1</span>],tre[rt].len,tre[rt].sum&#125;;</span><br><span class="line">		<span class="keyword">return</span>;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">int</span> mid = l + r &gt;&gt; <span class="number">1</span>;</span><br><span class="line">	<span class="keyword">if</span>(mid &gt;= R) modify(ls[rt], l, mid, L, R, B);</span><br><span class="line">	<span class="keyword">else</span> <span class="keyword">if</span>(mid &lt; L) modify(rs[rt], mid + <span class="number">1</span>, r, L, R, B);</span><br><span class="line">	<span class="keyword">else</span> modify(ls[rt], l, mid, L, mid, B),modify(rs[rt], mid + <span class="number">1</span>, r, mid + <span class="number">1</span>, R, B + mid - L + <span class="number">1</span>);</span><br><span class="line">	tre[rt].sum = add(tre[rt].sum, Fib(B, R - L + B));</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">Query</span><span class="params">(<span class="keyword">int</span> rt,<span class="keyword">int</span> l,<span class="keyword">int</span> r,<span class="keyword">int</span> L,<span class="keyword">int</span> R)</span></span>&#123;</span><br><span class="line">	<span class="keyword">if</span>(!rt) <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">	<span class="keyword">int</span> cur = Sib(tre[rt],L,R,l);</span><br><span class="line">	<span class="keyword">if</span>(l == L &amp;&amp; r == R) <span class="keyword">return</span> add(tre[rt].sum, cur);</span><br><span class="line">	<span class="keyword">int</span> mid = l + r &gt;&gt; <span class="number">1</span>;</span><br><span class="line">	<span class="keyword">if</span>(mid &gt;= R) <span class="keyword">return</span> add(Query(ls[rt], l, mid, L, R), cur);</span><br><span class="line">	<span class="keyword">else</span> <span class="keyword">if</span>(mid &lt; L) <span class="keyword">return</span> add(Query(rs[rt], mid + <span class="number">1</span>, r, L, R), cur);</span><br><span class="line">	<span class="keyword">else</span> <span class="keyword">return</span> add(add(Query(ls[rt], l, mid, L, mid), Query(rs[rt], mid + <span class="number">1</span>, r, mid + <span class="number">1</span>, R)), cur);</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	freopen(<span class="string">"you.in"</span>, <span class="string">"r"</span>, <span class="built_in">stdin</span>);</span><br><span class="line">	freopen(<span class="string">"you.out"</span>, <span class="string">"w"</span>, <span class="built_in">stdout</span>);</span><br><span class="line">	<span class="keyword">int</span> n = read(), m = read();</span><br><span class="line">	<span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i &lt;= n; ++i)</span><br><span class="line">		sum[i] = add(sum[i - <span class="number">1</span>], read() % mod);</span><br><span class="line">	init(n);</span><br><span class="line">	<span class="keyword">int</span> opt, l, r;</span><br><span class="line">	<span class="keyword">int</span> root = <span class="number">0</span>;</span><br><span class="line">	<span class="keyword">while</span>(m--)&#123;</span><br><span class="line">		opt = read() - <span class="number">1</span>, l = read(), r = read();</span><br><span class="line">		<span class="keyword">if</span>(!opt) &#123;</span><br><span class="line">			modify(root, <span class="number">1</span>, n, l, r, <span class="number">1</span>);</span><br><span class="line">		&#125;</span><br><span class="line">		<span class="keyword">else</span> <span class="built_in">printf</span>(<span class="string">"%d\n"</span>,add(Sum(l,r), Query(<span class="number">1</span>, <span class="number">1</span>, n, l, r)));</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

      
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